PhD Opportunity:

Modelling Ancient Geometric Reasoning


Type of Opportunity

PhD Project (1 studentship)

Advertising services

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Application deadline date

14th January 2019

Institution

University of Birmingham

https://www.birmingham.ac.uk/

School

School of Computer Science

https://www.cs.bham.ac.uk/

Supervisor(s)

Professor Aaron Sloman (contact for enquiries)

   https://www.cs.bham.ac.uk/~axs/
   a.sloman at cs.bham.ac.uk

Professor Achim Jung

  https://www.cs.bham.ac.uk/~axj/


Project title


Can current AI reasoning mechanisms be used to model ancient geometric reasoning, illustrating Immanuel Kant's ideas about mathematical knowledge?

Funding availability

Directly funded PhD project
(UK/EU students only)


Name of funding awarded


School of Computer Science Research Studentship/
School of Computer Science Teaching Assistantship

Funding notes

Details of funding available and eligibility requirements.
See Advert
Note: the studentship is advertised as part time because the recipient will be expected to help with teaching in the School of Computer Science.


Project description

This project is supervised by a theoretical computer scientist (Jung) and a philosopher of mathematics who has worked in AI and Cognitive Science (Sloman).

Immanuel Kant's Critique of Pure Reason (1781) claimed that mathematical knowledge about necessary truths and impossibilities differed from empirical knowledge (which requires extensive testing in different contexts) and also differed from truths derived logically from definitions (analytic knowledge).

Many 20th Century thinkers believed that Kant was proved wrong, e.g. because Einstein's general theory of relativity, supported by Eddington's observations of a 1919 eclipse of the sun showing apparent displacement of distant stars seen through the sun's gravitational field, showed that Euclidean geometry could be refuted empirically, while use of logic and David Hilbert's axiomatisation of Euclidean geometry, showed that truths of Euclidean geometry were analytic.

Sloman's 1962 DPhil thesis, recently digitised, http://www.cs.bham.ac.uk/research/projects/cogaff/sloman-1962/ , was an attempt to defend Kant.

A much stronger defence of Kant could come from an AI model of ancient mathematical thinking (e.g. thinking done by Archimedes, Euclid, Pythagoras, Zeno, etc.) But so far all the automated geometrical theorem provers, starting from Gelernter's 1964 theorem prover

https://pdfs.semanticscholar.org/2edc/8083073837564306943aab77d6dcc19d0cdc.pdf
are based on logic and the Cartesian coordinate representation of geometry.

Is it possible to use AI techniques to replicate the ancient modes of geometric discovery? Or is that beyond the scope of digital computer programs?

This project will investigate evidence collected so far, including the kinds of examples of mathematical discovery discussed in this invited lecture at an IJCAI workshop in 2017 http://www.cs.bham.ac.uk/research/projects/cogaff/misc/ijcai-2017-cog.html
Can new ideas about sub-neural, molecular mechanisms, combining discrete and continuous processes, be replicated and shown to be better able to model human geometric discovery processes and the spatial intelligence of squirrels, crows, elephants, and pre-verbal human toddlers?

E.g. see Trettenbrein 2016, The Demise of the Synapse As the Locus of Memory: A Looming Paradigm Shift?, Frontiers in Systems Neuroscience, Vol 88, http://doi.org/10.3389/fnsys.2016.00088

Why was Alan Turing working on chemical processes that include a mixture of continuous and discrete changes, shortly before he died (The Chemical Basis of Morphogenesis, published 1952)? Compare these papers discussing Turing's notion of mathematical intuition:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/turing-intuition-sloman.html)
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/super-turing-geom.html

This project has many facets that might suit students with different backgrounds, including mathematics, computer science, AI/Robotics, philosophy, psychology, neuroscience or biology, though previous experience of the use of diagrams in geometrical and topological reasoning and some AI programming experience will be particularly useful.

The research could be purely theoretical or could include development of a working model, or discuss the adequacy of current forms of computation for modelling ancient discovery processes. Or it could survey and criticise publications on the nature of mathematical knowledge e.g. whether it is innate.



Further References

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Research hours available


Full or part-time (This is the usual availability to allow widest accessibility to study)



Application link



https://sits.bham.ac.uk/lpages/EPS003.htm




Contacts for enquiries


Names of joint supervisors

Professor Achim Jung (main contact).
     http://www.cs.bham.ac.uk/~axj

Professor Aaron Sloman
     http://www.cs.bham.ac.uk/~axs

Email address

A.Jung@cs.bham.ac.uk

A.Sloman@cs.bham.ac.uk

Phone number

(+44) 121 41 44776